Almost isolated spectral parts and invariant subspaces
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- by C. R. Putnam
- Trans. Amer. Math. Soc. 216 (1976), 267-277
- DOI: https://doi.org/10.1090/S0002-9947-1976-0385599-7
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Abstract:
Let T be an operator with spectrum $\sigma (T)$ on a Hilbert space. A compact subset E of $\sigma (T)$ is called a disconnected part of $\sigma (T)$ if, for some bounded open set A, E is the closure of $\sigma (T) \cap A$ and $\sigma (T) - E$ is the union of the isolated parts of $\sigma (T)$ lying completely outside the closure of A. The set E is called an almost isolated part of $\sigma (T)$ if, in addition, the set A can be chosen so as to have a rectifiable boundary $\partial A$ on which the subset $\sigma (T) \cap \partial A$ has arc length measure 0. The following results are obtained. If T is subnormal and if E is a disconnected part of $\sigma (T)$ then there exists a reducing subspace $\mathfrak {M}$ of T for which $\sigma (T|\mathfrak {M}) = E$. If ${T^\ast }$ is hyponormal and if E is an almost isolated part of $\sigma (T)$ then there exists an invariant subspace $\mathfrak {M}$ of T for which $\sigma (T|\mathfrak {M}) = E$. An example is given showing that if T is arbitrary then an almost isolated part of $\sigma (T)$ need not be the spectrum of the restriction of T to any invariant subspace.References
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Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 216 (1976), 267-277
- MSC: Primary 47A10; Secondary 47B20
- DOI: https://doi.org/10.1090/S0002-9947-1976-0385599-7
- MathSciNet review: 0385599